integral from 0 to pi of sin(x)*cos(nx)

解答

\int_{0}^{π} sin (x) \cdot cos (n x) d x

- \frac{( - 1 ) ^{- n} + 1}{( n + 1 ) ( n - 1 )}

求解步骤

\int_{0}^{π} sin (x) cos (n x) d x

利用以下特性:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int_{0}^{π} \frac{sin ( x + n x ) + sin ( x - n x )}{2} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{0}^{π} sin (x + n x) + sin (x - n x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{0}^{π} sin (x + n x) d x + \int_{0}^{π} sin (x - n x) d x)

\int_{0}^{π} sin (x + n x) d x = \frac{( - 1 ) ^{n} + 1}{1 + n}

\int_{0}^{π} sin (x - n x) d x = \frac{( - 1 ) ^{- n} + 1}{1 - n}

= \frac{1}{2} (\frac{( - 1 ) ^{n} + 1}{1 + n} + \frac{( - 1 ) ^{- n} + 1}{1 - n})

化简

\frac{1}{2} (\frac{( - 1 ) ^{n} + 1}{1 + n} + \frac{( - 1 ) ^{- n} + 1}{1 - n}) : - \frac{( - 1 ) ^{- n} + 1}{( n + 1 ) ( n - 1 )}

= - \frac{( - 1 ) ^{- n} + 1}{( n + 1 ) ( n - 1 )}

\int_{- 1}^{4} x^{2} - 3 x - 4 d x

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